On the zeros of a class of polynomials including the generalized Bessel polynomials (Q1318393)

Using the tridiagonal background of the defining equations for the polynomials \[ a_{n+1} P_{n+1}(x)- a_ n P_{n-1}(x)- b_ n P_ n(x)= c_ n xP_ n(x);\;P_ 0(x)=0,\;P_ 1(x)= 1, \] (called `a finite Hilbert-space setting' by the authors) a very short and elegant proof is given of the main result: if \(b_ n>0\), \(a_ n\neq 0\), \(c_ n>0\) then every zero of \(P_ n\) has negative real part. Applications to the situation of associated Bessel polynomials (\(a_ n=1;\;b_ n=\delta_{n,1};\;c_ n=2(n+\nu)\)-including the Bessel polynomials for \(\nu= 1/2\)) and the generalized Bessel polynomials are given, reproving and (partially) improving known results. Also bounds for the (negative) real parts of the zeros are given.